Line Spectrum of Hydrogen

James Richard Fromm

The continuous spectrum of black-body radiation is emitted by all objects heated to high temperatures or otherwise supplied with sufficient energy -- a process called excitation. Any object which is not a perfect radiation source, or ideal black body, is also found to emit strongly at specific wavelengths in addition to its continuous spectrum emission. These specific wavelengths, which are sharply defined, are highly characteristic of the substance being excited; they are called line spectra. Isolated atoms, which are the form of matter found at extremely high temperatures, emit line spectra which are characteristic of the element whose atoms are being excited. Chemists make use of this to identify the elements present in samples; the technique is called emission spectroscopy.

In emission spectroscopy, a source of energy, usually heat as in a flame or an electric arc or spark, is passed into the sample, which absorbs such frequencies as are possible; the re-emission of this light by the sample is observed. The process of absorption and re-emission produces a bright line spectrum which can be measured when the light is dispersed by a grating or prism. The wavelengths of the emitted light identify the element and the particular electronic transition occurring within it. The amount of light emitted is related to the amount of the element present, and if sufficient precautions are taken the amount can be semiquantitatively determined by measuring the amount of this light. This can be done photographically or by more complex electronic procedures.

In absorption spectroscopy, light of a particular single frequency is passed through a sample which must be in solution or in gaseous form. The amount of light absorbed is directly relatable to the amount of the particular species present to absorb it. Conditions are somewhat easier to control in this arrangement and quantitative (1%) precision can be obtained. The use of absorption spectroscopy is more general for the lower-energy vibrational and rotational transitions but can be used for electronic transitions as well, as is done in atomic absorption spectroscopy.

When cooler samples of various substances are placed between a black body at a higher temperature and an observer, the observer will see a dark line spectrum superimposed on the continuous spectrum of the black body. This demonstrates that substances absorb light at discrete wavelengths, and it is observed that these wavelengths at which substances absorb light are the same wavelengths as those at which they emit light. The element helium was discovered on the sun through its dark line spectrum in this way before it was discovered on earth.

The Spectrum of Hydrogen

The line spectrum of the simplest element, hydrogen, was studied well before the turn of the century and the wavelengths of its lines were well known. In the visible range, a series of lines is observed at the following wavelengths (in nm): 656.279, 486.133, 434.047, 410.174, 397.007, 388.905, 383.539, and 379.790. The series continues to shorter wavelengths, the difference between successive wavelengths continually decreasing. Similar patterns were found to exist in the line spectra of many elements.

A wide variety of empirical formulae were used to correlate these wavelengths in the attempt to understand the nature of the processes causing them. One of the most successful was that of the Swiss physicist Balmer, who in 1885 showed that the wavelength l of each member of this particular series of hydrogen lines were very accurately given by what came to be known as the Balmer equation:

l = 364.56(a2/(a2 - 22))

In the Balmer equation, which in this form gives the wavelength lin nm, a is an integer having the values 3, 4, 5, 6,.... Each value of an integer corresponded to an observed wavelength. This series of lines became known as the Balmer series.

Additional series of lines were later discovered for hydrogen in the ultraviolet region of the spectrum (Lyman series, 1908) and in the infrared region of the spectrum (Paschen series, Brackett series, and Pfund series). A more general formula proposed by Ritz which included all four of these series is, in comparable form:

l = 91.14k2(a2/(a2 - k2))

In this equation k is the integer 1 (Lyman), 2 (Balmer), 3 (Paschen), 4 (Brackett), 5 (Pfund), ... and a is an integer ranging upwards from k + 1. This equation, the Rydberg equation, is more commonly written in the form

1/l = RH(a2 - k2)/a2k2

The constant RH is the Rydberg constant for hydrogen and is one of the most accurately measured fundamental constants of the universe. It has the value 10,967,758.1 +/- 0.7 m-1.

Not only does the Rydberg equation give very accurate results for the hydrogen atom, with some modification it represents accurately the line spectrum of many other atoms. Since the spectrum of an atom must be related in some way to its structure, the structure of atoms must be related to a series of integers. Until 1913, scientists had not been able to devise any structure or model which accounted for this embarrassing experimental fact.

Line Spectrum and Energy Structure

The energy structure of the hydrogen atom can be calculated directly from the Rydberg equation. Since lf = c, f = cRH(a2 - k2)/a2k2 gives the observed frequencies, which can be expressed as energies using the Planck equation E(f = hf:

E(f) = hcRH(a2 - k2)/a2k2

For the lowest-energy line in the Lyman series, for which k = 1 and a = 2, the energy emitted as a single quantum (single photon) by a single atom is

E(f) = hcRH(4 - 1)/4 = 3hcRH/4

The value for one mole of atoms is that for a single atom multiplied by the Avogadro number, E(f) = 3hcRHNA/4, so

E(f) = 3(6.6260755 x 10-34 J s)(299792458 m/s)(10967758.1 m-1)(6.0221367 x 10+23 mol-1)/4

E(f) = 984.027 kJ/mol

The remaining lines of the Lyman series, the series for which k = 1, are:

E(a = 2) = 984.027 kJ/mol (3/4)
E(a = 3) = 1166.254 kJ/mol (8/9)
E(a = 4) = 1230.034 kJ/mol (15/16)
E(a = 5) = 1259.554 kJ/mol (24/25)
E(a = 6) = 1275.591 kJ/mol (35/36), ... to
E(a = inf.) = 1312.036 kJ/mol (1)

In the previous section it was indicated that the interaction of light with matter gives information related to the structure of matter. In order to understand this interaction, it is necessary to realize that light is not a continuous wave of some particular frequency, as physicists of the previous century thought. Instead, we now view light as a succession of packets of energy in wave form. These "waveicles" are called quanta or photons. Each individual atomic or molecular event gives rise to one and only one photon. A stream of photons of the same energy, which is a beam of light of a single frequency, can arise only from a large number of identical atoms or molecules which are continually emitting photons due to identical events occurring within them.

Measurement of the energy of one quantum of light, or one photon, is a simple calculation. Frequencies or wavelengths can be directly converted into energy using the Planck equation E(f = hf. For any frequency of light f the Planck equation permits calculation of the energy change corresponding to its emission or absorption. For a frequency of 3 x 10+14 Hz, corresponding to a wavelengths of 1000 nm, that would be about 20 x 10-20 J, rather a small number. This value is joules per atom or molecule, since the light was emitted or absorbed by only one atom or molecule at a time. To compare this value with the energies of chemical bonds we must multiply by the number of molecules per mole, which is Avogadro's number NA where NA = 6.02252 0.00028 x 10+23 entities/mole.

This energy is about 20 x 10-20 x 6 x 10+23 = 120,000 J/mole or 120 kJ/mole for absorption or emission of light at 1000 nm. Most spectrometers dealing with electronic energy transitions operate in the 200 - 800 nm range where the energies involved are more like 150 - 600 kJ/mole.

Now let us compare this energy with that in some typical chemical bonds involving hydrogen. An average carbon-hydrogen single bond has a bond energy of 414 kJ/mole, an average hydrogen-hydrogen single bond has a bond energy of 435 kJ/mole, and an average oxygen-hydrogen single bond has a bond energy of 464 kJ/mole. Clearly, the electronic energy for hydrogen excitation, as measured spectroscopically, is greater than that of the bonds between a hydrogen atom and any other atom. In general, this is true for all atoms. As a consequence, electronic energy changes cannot give us much information about chemical bonding because the energy involved in them is sufficient to rupture the bonds. They can, however, give us a great deal of information about the types and relative proportions of the atoms present in a particular sample, and they are used in practical chemical analysis for this purpose.

Copyright 1997 James R. Fromm